Sounds like a poor ground, when the headlights are off then the 21W can get to ground via the headlamps but if the headlamps are on then that does not happen.
Check the ground connections and for breaks in the ground wires.
Without a turn signal on coming drivers would not know if you are wanting to make a right/left hand turn. This would reduce safety if turn signals were not a thing. In other words its a form of communication with other users on the road.
Today I approached a give-way at a more major (but not very wide) road, and noticed a bus approaching. Because it was indicating to turn my way, I realised that I should stop short of the give-way line to allow it room to turn across the front of me. If I had stopped right at the line, we would have needed to reverse the priorities, which can be confusing ...
What Mike said and also:
On the highway; letting drivers in front of you know that you're switching lanes. That way they know that you're about to overtake them
When parked, oncoming drivers as well as drivers parked in front of you know you're about to drive off, so they know not to overtake or drive off.
When leaving a roundabout, oncoming drivers know ...
So drivers travelling in the opposite direction know you have an intention to cross their lane.
You only need to check youtube to see drivers who don’t see cars in other lanes or those who don’t use the indicators (don’t know how to use, don’t know they are fitted...)
I had a brake light problem like this before where it was on the whole time and it went unnoticed. It eventually drained my battery lol. The cause was a broken brake light switch stopper which can be found under your pedals. Better check that out as well.